Bayesian inference of Weibull distribution for right and interval censored data
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(2) H. By. T. U. PM. BAYESIAN INFERENCE OF WEIBULL DISTRIBUTION FOR RIGHT AND INTERVAL CENSORED DATA. ©. C. O. PY. R. IG. CHRIS BAMBEY GUURE. Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia, in Fulfilment of the Requirements for the Degree of Master of Science. June 2013.
(3) COPYRIGHT All material contained within the thesis, including without limitation text, logos, icons, photographs and all other artwork, is copyright material of Universiti. PM. Putra Malaysia unless otherwise stated. Use may be made of any material contained within the thesis for non-commercial purposes from the copyright holder.. Commercial use of material may only be made with the express, prior, written. U. permission of Universiti Putra Malaysia.. ©. C. O. PY. R. IG. H. T. Copyright c Universiti Putra Malaysia.
(4) DEDICATION. ©. C. O. PY. R. IG. H. T. U. PM. To Syntyche Guure. i.
(5) Abstract of thesis presented to the Senate of Universiti Putra Malaysia in fulfilment of the requirement for the degree of Master of Science. BAYESIAN INFERENCE OF WEIBULL DISTRIBUTION FOR RIGHT AND INTERVAL CENSORED DATA. CHRIS BAMBEY GUURE. T. Chair: Professor Noor Akma Ibrahim, Ph.D.. U. June 2013. PM. By. H. Faculty: Institute for Mathematical Research. IG. The main purpose of this work is to draw comparisons between the classical maximum likelihood and the Bayesian estimators on the parameters, the. R. survival function and hazard rate of the Weibull distribution when the data. PY. under consideration are right and interval censored. We have considered the survival data to follow Weibull distribution due to its adaptability in fitting time-to-failure of a very widespread multiplicity to multifaceted mechanisms. O. in the field of life-testing and survival analysis.. C. In Bayesian estimations, prior distributions as well as loss functions need to. ©. be specified. The prior distributions can be obtained via previous study in relation to the current study or by soliciting information from experts. We have considered in this study, different types of priors, such as, Jeffreys prior, extension of Jeffreys’ prior information, gamma priors and have also proposed a generalised non-informative prior. The loss functions considered in this study are asymmetric and symmetric loss functions.. ii.
(6) Lindley’s approximation procedure is used in the Bayesian estimation approach to reduce the ratio of integrals in the posterior distributions which cannot be obtained in close forms. When we consider both the scale and shape parameters under the right and interval censored data, we observed that the estimate of the shape parameter under the maximum likelihood method can-. PM. not be obtained in close form; therefore, a numerical approach known as Newton-Raphson has been employed to estimate the shape parameter.. U. The mean squared errors and mean absolute biases of the estimates under Bayes and its maximum likelihood counterpart are examined through sim-. T. ulation study under several conditions to evaluate the performance of both methods. Overall, it has been observed that, the proposed Bayesian estima-. H. tion under the generalised non-informative prior performed better than the. IG. other estimators for the scale and shape parameters, the survival function and hazard rate. The Bayesian estimator via the generalised non-informative prior. R. occurred largely with the linear exponential loss function followed by general. ©. C. O. PY. entropy loss function.. iii.
(7) Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai memenuhi keperluan untuk ijazah Master Sains. Oleh CHRIS BAMBEY GUURE. T. Pengerusi: Profesor Noor Akma Ibrahim, Ph.D.. U. Jun 2013. PM. PENTAKBIRAN BAYES BAGI TABURAN WEIBULL UNTUK DATA MANDIRAN TERTAPIS KEKANAN DAN SELANG. IG. H. Fakulti: Institut Penyelidikan Matematik. Tujuan utama kajian ini adalah untuk membuat perbandingan di antara kebolehjadian maksimum klasik dengan penganggar Bayesian terhadap param-. R. eter, fungsi mandirian dan kadar bahaya taburan Weibull apabila data di. PY. bawah pertimbangan adalah tertapis kekanan dan tertapis berselang. Kami telah mempertimbangkan data mandirian bertaburan Weibull kerana kebolehsuaiannya dalam suaian masa-terhadap-kegagalan dalam mekanisme pelbagai. O. aspek dalam bidang ujian hayat dan analisis mandirian.. C. Dalam penganggaran Bayes, taburan prior serta fungsi kehilangan perlu diny-. ©. atakan. Taburan prior boleh diperolehi melalui kajian sebelumnya berhubung dengan kajian semasa atau dengan meminta maklumat daripada pakar. Kami telah mempertimbangkan dalam kajian ini, jenis prior yang berbeza, seperti prior Jeffreys, lanjutan kepada prior Jeffreys, prior gamma dan juga telah mencadangkan prior tak bermaklumat teritlak. Fungsi kehilangan yang dipertimbangkan dalam kajian ini adalah asimetri dan fungsi kehilangan simetri.. iv.
(8) Tatacara penghampiran Lindley digunakan dalam pendekatan penganggaran Bayes untuk mengurangkan nisbah kamiran dalam taburan posterior yang tidak boleh diperolehi dalam bentuk mudah. Apabila kita mempertimbangkan parameter skala dan parameter bentuk bagi data tertapis kekanan dan tertapis berselang, kita perhatikan bahawa anggaran parameter bentuk yang di bawah. PM. kaedah kebolehjadian maksimum tidak boleh boleh diperolehi dalam bentuk. mudah. Oleh itu, satu pendekatan berangka yang dikenali sebagai Newton-. U. Raphson telah digunakan untuk menganggar parameter bentuk.. Purata ralat kuasa dua dan kepincangan mutlak bagi penganggar di bawah. T. Bayes dan kebolehjadian maksimum diteliti melalui kajian simulasi di bawah beberapa syarat untuk menilai prestasi kedua-dua kaedah.. Dapat diper-. H. hatikan bahawa prior tak bermaklumat teritlak yang dicadangkan bagi Bayes. IG. menunjukkan yang prestasinya adalah lebih baik daripada penganggar kaedah lain bagi parameter skala dan parameter bentuk, fungsi mandirian dan kadar. R. bahaya. Ini berlaku kebanyakannya dengan fungsi kehilangan linear ekspo-. ©. C. O. PY. nen diikuti oleh dengan fungsi kehilangan entropi teritlak.. v.
(9) ACKNOWLEDGEMENTS. Principally, my utmost and sincerest gratitude is to God Almighty for having blessed me from the start of the write-up to this very end. I am highly and. PM. sincerely grateful to Professor Dr. Noor Akma Ibrahim, for her incredible and exceptional guidance, the excellent and fruitful discussions, useful directions and that of her tremendous and continuous support. It has been an honour. U. working with her. May I also take this opportunity to thank Dr. Mohd Bakri Adam, for his creative guidance, comments, contributions, suggestions and. T. encouragement. They have been very helpful.. H. My earnest gratitude is to my entire family members for their prayers, sup-. IG. port, motivation and encouragement towards this triumphant success. Last but not the least is to my lovely wife Grace Kambey Guure for her prayers. ©. C. O. PY. R. and support.. vi.
(10) ©. T. H. IG. R. PY. O. C. PM. U.
(11) This thesis was submitted to the Senate of Universiti Putra Malaysia and has been accepted as a fulfilment of the requirement for the degree of Master of Science. The members of the Supervisory Committee were as follows:. PM. Noor Akma Ibrahim, Ph.D. Professor Institute for Mathematical Research Universiti Putra Malaysia (Chairperson). ©. C. O. PY. R. IG. H. T. U. Mohd Bakri Adam, Ph.D. Senior Lecturer Faculty of Science Universiti Putra Malaysia (Member). BUJANG BIN KIM HUAT, Ph.D. Professor and Dean School of Graduate Studies Universiti Putra Malaysia Date: viii.
(12) DECLARATION. ©. C. O. PY. R. IG. H. T. U. PM. I declare that the thesis is my original work except for quotations and citations which have been duly acknowledged. I also declare that it has not been previously, and is not concurrently, submitted for any other degree at Universiti Putra Malaysia or at any other institution.. CHRIS BAMBEY GUURE Date: 13 June 2013. ix.
(13) TABLE OF CONTENTS. Page i ii iv vi vii ix xii xv. U. PM. DEDICATION ABSTRACT ABSTRAK ACKNOWLEDGEMENTS APPROVAL DECLARATION LIST OF TABLES LIST OF ABBREVIATIONS. BAYESIAN ESTIMATE OF RELIABILITY AND HAZARD RATE 3.1 Introduction 3.2 Maximum Likelihood Estimation 3.3 Bayesian Inference 3.3.1 Bivariate Prior 3.3.2 Generalised non-informative Prior 3.3.3 Linear Exponential Loss Function 3.3.4 Lindley Approximation 3.3.5 General Entropy Loss Function 3.3.6 Symmetric Loss Function 3.4 Simulation Study 3.5 Results and Discussion 3.6 Conclusion. ©. C. 3. 1 1 5 6 7 8 9 10 12 13 13 16 16 24 26. LITERATURE REVIEW 2.1 Background 2.2 The Loss Functions 2.3 Prior Distribution. O. 2. PY. R. IG. H. T. CHAPTER 1 INTRODUCTION 1.1 Background 1.2 Censoring 1.2.1 Right Censoring 1.2.2 Interval Censoring 1.3 Survival/Reliability Function 1.4 Hazard/Failure Rate 1.5 Weibull Distribution 1.6 Problem Statement 1.7 Objectives 1.8 Delineation. x. 29 29 29 32 32 33 35 36 38 40 42 43 53.
(14) BAYESIAN ESTIMATE WITH EXTENSION OF JEFFREYS’ PRIOR 4.1 Introduction 4.2 Maximum Likelihood Estimation 4.3 Bayesian Estimation 4.4 The Loss Functions 4.4.1 Linear Exponential Loss Function 4.4.2 General Entropy Loss Function 4.4.3 Symmetric Loss Function 4.5 Simulation Study 4.6 Results and Discussion 4.7 Conclusion. 54 54 54 55 56 56 57 58 59 59 65. 5. BAYESIAN ANALYSIS OF THE SURVIVAL AND FAILURE RATE 5.1 Introduction 5.2 Maximum Likelihood Estimation 5.3 Bayesian Methods 5.3.1 Informative Prior 5.3.2 Generalised Non-Information Prior 5.3.3 Linear Exponential Loss Function 5.3.4 Lindley Approximation 5.3.5 General Entropy Loss Function 5.3.6 Squared Error Loss Function 5.4 Simulation Study 5.5 Results and Discussion 5.6 Conclusion. 66 66 66 68 69 70 70 71 73 75 76 78 94. 6. BAYESIAN INFERENCE BASED ON INTERVAL CENSORING 6.1 Introduction 6.2 Maximum Likelihood Estimation 6.3 Bayesian Analysis 6.3.1 Informative Prior 6.3.2 Lindley Approximation 6.3.3 Linear Exponential Loss Function 6.3.4 General Entropy Loss Function 6.4 Simulation Study 6.5 Results and Discussion 6.6 Real Data Analysis 6.7 Conclusion. 96 96 96 98 99 100 102 103 103 105 109 113. CONCLUSION AND FUTURE RESEARCH 7.1 Conclusion 7.2 Future Research. 114 114 116. ©. C. O. PY. R. IG. H. T. U. PM. 4. 7. 118 123 134 135. BIBLIOGRAPHY APPENDICES BIODATA OF STUDENT LIST OF PUBLICATIONS xi.
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